Hyperloop

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The Hyperloop is a proposed high-speed transportation system in which specialized pods are accelerated through a low-pressure tube to achieve speeds near the speed of sound. The system is intended to provide a more cost-effective and faster mode of travel between cities separated by distances less than approximately nine hundred miles [1]. First conceptualized in 2012 by Elon Musk, the tech entrepreneur leading numerous high-profile companies including SpaceX and Tesla Motors, hypothetical benefits of the Hyperloop include immunity to weather, lack of crashes, rapid speed, low power requirements, and energy generation for its own operation [2].

Artist's concept of an operating Hyperloop; image taken from [3].

Designing and building the Hyperloop poses numerous technical and economic challenges. As of early 2016, a group of approximately one hundred engineers from across the United States are completing a technical feasibility study to assess if construction is realistic from an engineering standpoint [2]. Musk has estimated that implementing a full passenger-plus-cargo version of the Hyperloop would cost approximately $7.5 billion USD, although critics claim that the final cost could end up being around ten times as much [1,4].

Contents

From Concept to Design

After Musk's initial proposal in 2012, engineers from both SpaceX and Tesla Motors worked informally for about nine months on creating a reasonable engineering and cost-assessment proposal for the Hyperloop, which was released in August 2013 [5].

Concept art of the Hyperloop pod system taken from Musk's "alpha" proposal of August 2013 [1].

Musk has described Hyperloop operation as a cross between the Concorde supersonic jet, a railgun (which can accelerate projectiles to supersonic speeds rapidly using the principles of electrodynamics), and an air hockey table, which creates small bearings of air that act as a cushion for objects to move with little friction [6]. In his original proposal, Musk suggests a Los Angeles - San Francisco route which could be traversed in about 35 minutes, in comparison with over five hours by car and over an hour by plane [2]. Energy to accelerate and maintain the speed of pods in the Hyperloop would be obtained from solar panels mounted along the track, as illustrated below:

Arrays of solar panels mounted on the tube for the Hyperloop capsules would provide the energy to sustain operation. Image taken from [1].

As of 2016, two major competing companies exist with separate proposals to bring the concept desing of the Hyperloop into reality. The first is Hyperloop Technologies, a "classic tech startup, with $37 million in venture funds, an industrial-chic office and 72 employees". The second, Hyperloop Transportation Technologies, is a worldwide consortium of 450 scientists, engineers, and entrepreneurs who work part-time in exchange for equity in the company [4]. Each is exploring different routes for their proposed Hyperloop tracks; the first is investigating a possible Los Angeles - Las Vegas route while the second has announced plans to build a partial track on the Los Angeles - San Francisco route from the original proposal [7]. Representatives from both companies, however, have suggested that it is more likely that a country in Asia or the Middle East implements a Hyperloop before the United States due to the reduced bureaucracy and increased need for efficient transportation in those regions [4].

Simultaneously, Musk's company SpaceX announced "an open competition, geared towards university students and independent engineering teams, to design and build the best Hyperloop pod" [8]. Over 120 teams entered this competition, the goal of which was to select a finalist(s) to build and test pods at SpaceX headquarters in Summer 2016, ideally to select a final design for a commercial Hyperloop [9]. On January 30, 2016, a team of graduate students from MIT won the first stage of this competition with an innovative pod design that uses magnetic levitation rather than an air cushion to reduce friction on pods [10].

Physics of the Hyperloop

Design of a realistic Hyperloop must surmount numerous technical challenges to achieve the desired speeds in a cost-effective way without risk to human health. This section summarizes solutions to these challenges as presented by Musk in his August 2013 "alpha" proposal of the Hyperloop.

To achieve Musk's proposed 35 minute trip between Los Angeles and San Francisco, a series of linear induction motors would accelerate a Hyperloop pod up to approximately 760 miles per hour, just below the speed of sound.

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The initial primary proposed route is from San Francisco to Los Angeles, a distance of \(560~\mbox{km}\). The Hyperloop is hypothesized to cover this distance in half an hour. If it does this at a constant speed, how fast is the speed of the Hyperloop in m/s?

In order for the pods to travel this quickly, there must be some way to alleviate friction forces between the pod and the tube. One possible mechanism is magnetic levitation: using very large permanent magnets to suspend the pod above the bottom of the tube. However, Musk dismissed this idea in his initial proposal due to cost [1]. A second possible method is akin to the air bearings employed by an air hockey table: using a cushion of air to support pods, eliminating rolling resistance.

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The Hyperloop reduces friction between the pods and the tunnel by supporting the pod on a cushion of air. If the mass of the Hyperloop pod is \(3100~\mbox{kg}\), what force in N must be exerted on the pod by the air cushion to keep it suspended?

The acceleration of gravity is \(-9.8~\mbox{m/s}^2\).

Another barrier to high-speed travel is the large friction force due to air resistance. One obvious solution is that pods could be propelled in a vacuum tube. However, the vacuum pumps required to achieve this would be extremely costly and any equipment malfunction could be devastating to Hyperloop operation. To address this problem, it is instead proposed that the Hyperloop tube operate at very low pressure: 100 Pascals, about 1/6 the pressure of the atmosphere of Mars [1]. This pressure is one thousand times less than atmospheric pressure at sea level and as a result air resistance is drastically decreased. After initial acceleration, Hyperloop pods can therefore mostly glide without applying any thrust until the deceleration at the end of the journey.

Model of air streamlines around a Hyperloop pod; image taken from [1].

Even though the Hyperloop tube will operate at very low pressure, pods must still be designed with aerodynamics in mind. To keep material costs down, the cross-sectional area of the tube would be very close to the cross-sectional area of the pods, so that the walls of the pod are close to the walls of the tube. If the walls are too close, the pod behaves as a syringe: it pushes the entire column of air in front of it, rather than letting the air flow around it. Since the tube would be hundreds of miles long, this effect would immensely throttle maximum pod speed. The top speed for a pod given a certain tube to pod cross-sectional area ratio is called the Kantrowitz limit [1].

To overcome this problem, the ideal pod will be engineered with an electric compressor fan on the nose of the pod. This will function to pump high-pressure air from in front of the pod to behind the pod. By pumping air underneath the pod, this would serve the dual purpose of providing the "air hockey table" air-cushion suspension effect [1].

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The tunnel and Hyperloop pod are both cylindrically symmetric. If the radius of the pod is too close to the radius of the tunnel, then the air flowing through the cracks will become supersonic, generating shockwaves and disrupting the smooth operation of the Hyperloop. The pod helps prevent this by sucking up and compressing 0.49 kg/s of air through the front of the pod as it travels, but this still means some air must flow around the pod.

If the radius of the tunnel is 1.1 m, the speed of the pod is 300 m/s and the air in the tunnel is at a pressure of 99 Pa and temperature of 293 K, then what is the maximum radius of the pod in m that will keep the air flow relative to the pod below the speed of sound?

Treat air as incompressible. This isn't true at high mach number, but it will make the solution easier.

The molar mass of air is \(\mu= 29\) g/mol.

Neglect the effects of gravity and viscosity.

Assume the air flow is perfectly laminar.

Musk's design proposes that passengers experience a maximum inertial acceleration of 0.5 g, only two to three times that of a passenger jet on takeoff and landing provided that the Hyperloop tube functions properly.

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The normal pressure in the tunnel is \(99~\mbox{Pa}\), which is very low compared to the usual atmospheric pressure of \(101,325~\mbox{Pa}\). Since the pressure is so low the Hyperloop tunnel must be well sealed to prevent outside air from rushing in. A sudden increase in air pressure in a section of the tunnel can be rather unpleasant for the passengers in the pod.

Consider for example a hole being created in the Hyperloop tunnel, which leads to a sudden increase in the local air pressure from \(99~\mbox{Pa}\) to \(101,325~\mbox{Pa}\) while maintaining constant temperature and volume. If the normal drag force on the Hyperloop is \(320~\mbox{N}\), how much acceleration in g's would the passengers in the Hyperloop experience if the pod hit the region of high pressure?

The mass of the pod is approximately \(15000~\mbox{kg}\).

\(g=9.8~\mbox{m/s}^2\)

Since the pod design for the Hyperloop is not yet finalized, the precise physics of the technologies that will be used to accelerate the pod, reduce friction, and address aerodynamic concerns are still subject to change.